Evaluate the Integral

Problem

(∫_0^1)((e(2*x)−e(−2*x))/(e(2*x)+e(−2*x))*d(x))

Solution

1. Identify the structure of the integrand and notice that the numerator is related to the derivative of the denominator.

2. Substitute u=e(2*x)+e(−2*x) to simplify the expression.

3. Differentiate u with respect to x to find d(u)

d(u)/d(x)=2*e(2*x)−2*e(−2*x)

d(u)=2*(e(2*x)−e(−2*x))*d(x)

1/2*d(u)=(e(2*x)−e(−2*x))*d(x)

4. Change the limits of integration from x to u
   When x=0

u=e0+e0=2

When x=1

u=e2+e(−2)

5. Rewrite the integral in terms of u

(∫_2^e2+e(−2))(1/(2*u)*d(u))

6. Integrate using the natural logarithm rule:

1/2*[ln(u)](e2+e(−2))2

7. Evaluate at the upper and lower limits:

1/2*(ln(e2+e(−2))−ln(2))

8. Simplify using the properties of logarithms:

1/2*ln((e2+e(−2))/2)

Final Answer

(∫_0^1)((e(2*x)−e(−2*x))/(e(2*x)+e(−2*x))*d(x))=1/2*ln((e2+e(−2))/2)

Want more problems? Check here!